Home›Calculus 1›Volumes by disks and washers›Problem 5.26

Volumes by disks and washers

Problem 5.26 · easy

The region under \( \displaystyle y = x^{2} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 4 \) is revolved about the x-axis. Find the volume.

Cross sections perpendicular to the x-axis are disks of radius f(x), with area πf(x)².
Reviewed
\[ \int\limits_{0}^{4} \pi x^{4}\, dx = \frac{1024 \pi}{5} \]
V = ∫ A(x) dx.✓ Proved

Answer \( \frac{1024 \pi}{5} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature of the cross-sectional area agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the disk method for rotation about the x-axis. The radius is correctly identified as f(x) = x^2, leading to the area function π(x^2)^2 = πx^4. The integration bounds and result are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the disk method for rotation about the x-axis. The radius is correctly identified as f(x) = x^2, leading to the area function π(x^2)^2 = πx^4. The integration bounds and result are correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the disk method for rotation about the x-axis, identifying the radius as f(x) and setting up the integral with the correct bounds and integrand.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the method of disks for rotation about the x-axis, sets up the integral with the correct radius and bounds, and obtains the correct result.
gpt-oss:20b: pass 2026-09-26

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/volume_disk_washer, checked 2026-09-28 with SymPy 1.14.0.